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Double Field Theory : Stringy Extension of General
Relativity
– ‘Uroboros’ solution to the Dark Matter problem –
Jeong-Hyuck Park 박정혁 (朴廷爀) Sogang University
9th March 2017, Workshop on GEOMETRY, DUALITY AND STRINGS @
YITP, Kyoto University
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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Prologue
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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General Relativity, Riemannian Geometry & String Theory
• Ever since Einstein formulated his theory of gravity i.e. GR,
by employing themathematics of Riemannian geometry, the Riemannian
metric, gµν , has beenprivileged to be the only geometric and hence
gravitational field:
– Diffeomorphism : ∂µ −→ ∇µ = ∂µ + Γµ
– ∇λgµν = 0, Γλ[µν] = 0 −→ Γλµν =
12 gλρ(∂µgνρ + ∂νgµρ − ∂ρgµν)
– Curvature : [∇µ,∇ν ] −→ Rκλµν −→ R
• On the other hand, string theory puts the metric, gµν ,
two-form gauge potential, Bµν ,and scalar dilaton, φ, on an equal
footing, as they, so called the massless NS-NS sector,form a
‘multiplet of T-duality’ (this string theory symmetry mixes
them).
• Namely, string theory suggests to view the whole massless
NS-NS sector as thegravitational unity.
– Riemannian geometry is for particle theory.– String theory
requires a novel differential geometry for the NS-NS sector.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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General Relativity, Riemannian Geometry & String Theory
• Ever since Einstein formulated his theory of gravity i.e. GR,
by employing themathematics of Riemannian geometry, the Riemannian
metric, gµν , has beenprivileged to be the only geometric and hence
gravitational field:
– Diffeomorphism : ∂µ −→ ∇µ = ∂µ + Γµ
– ∇λgµν = 0, Γλ[µν] = 0 −→ Γλµν =
12 gλρ(∂µgνρ + ∂νgµρ − ∂ρgµν)
– Curvature : [∇µ,∇ν ] −→ Rκλµν −→ R
• On the other hand, string theory puts the metric, gµν ,
two-form gauge potential, Bµν ,and scalar dilaton, φ, on an equal
footing, as they, so called the massless NS-NS sector,form a
‘multiplet of T-duality’ (this string theory symmetry mixes
them).
• Namely, string theory suggests to view the whole massless
NS-NS sector as thegravitational unity.
– Riemannian geometry is for particle theory.– String theory
requires a novel differential geometry for the NS-NS sector.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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General Relativity, Riemannian Geometry & String Theory
• Ever since Einstein formulated his theory of gravity i.e. GR,
by employing themathematics of Riemannian geometry, the Riemannian
metric, gµν , has beenprivileged to be the only geometric and hence
gravitational field:
– Diffeomorphism : ∂µ −→ ∇µ = ∂µ + Γµ
– ∇λgµν = 0, Γλ[µν] = 0 −→ Γλµν =
12 gλρ(∂µgνρ + ∂νgµρ − ∂ρgµν)
– Curvature : [∇µ,∇ν ] −→ Rκλµν −→ R
• On the other hand, string theory puts the metric, gµν ,
two-form gauge potential, Bµν ,and scalar dilaton, φ, on an equal
footing, as they, so called the massless NS-NS sector,form a
‘multiplet of T-duality’ (this string theory symmetry mixes
them).
• Namely, string theory suggests to view the whole massless
NS-NS sector as thegravitational unity.
– Riemannian geometry is for particle theory.– String theory
requires a novel differential geometry for the NS-NS sector.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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General Relativity, Riemannian Geometry & String Theory
• However, in the conventional treatment of the NS-NS sector,
the effective actiondescribing its dynamics is ‘organized’ in terms
of Riemannian geometry,
∫dDx
√−|g| e−2φ
(R + 4 |dφ|2 − 112 |dB|
2).
– In this conventional description, the Riemannian metric
provides the backgroundgeometry, while the dilaton and the B-field
are viewed as ‘matter’ living on it.
– Further, the O(D,D) T-duality symmetry mixing the NS-NS sector
is notmanifest at all, while it is secretly hidden there.
– There is also much ambiguity to occur, when we try to couple
the NS-NS sector,especially φ and Bµν , to other matters, e.g. the
Standard Model.
• Thus, Riemannian geometry fails to provide the unifying
geometric description of themassless NS-NS sector.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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Dark Matter Problem
Galaxy rotation curves : observation Keplerain 1/√
R fall-off : GR or Newton
• The galaxy rotation curve is a plot of the orbital velocities
of visible stars versus theirradial distance from the galactic
center.
• While Einstein gravity (GR), with Schwarzschild solution,
predicts the Keplerian(inverse square root) monotonic fall-off of
the velocities, observations however showrather ‘flat’ (∼ 200 km/s)
curves after a fairly rapid rise.
• The resolution of the discrepancy may call for ‘dark matter’,
or modifications of thelaw of gravity, or perhaps both as is the
case with Double Field Theory.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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Dark Matter Problem
Galaxy rotation curves : observation Keplerain 1/√
R fall-off : GR or Newton
• The galaxy rotation curve is a plot of the orbital velocities
of visible stars versus theirradial distance from the galactic
center.
• While Einstein gravity (GR), with Schwarzschild solution,
predicts the Keplerian(inverse square root) monotonic fall-off of
the velocities, observations however showrather ‘flat’ (∼ 200 km/s)
curves after a fairly rapid rise.
• The resolution of the discrepancy may call for ‘dark matter’,
or modifications of thelaw of gravity, or perhaps both as is the
case with Double Field Theory.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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Dark Matter Problem
Galaxy rotation curves : observation Keplerain 1/√
R fall-off : GR or Newton
• The galaxy rotation curve is a plot of the orbital velocities
of visible stars versus theirradial distance from the galactic
center.
• While Einstein gravity (GR), with Schwarzschild solution,
predicts the Keplerian(inverse square root) monotonic fall-off of
the velocities, observations however showrather ‘flat’ (∼ 200 km/s)
curves after a fairly rapid rise.
• The resolution of the discrepancy may call for ‘dark matter’,
or modifications of thelaw of gravity, or perhaps both as is the
case with Double Field Theory.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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Talk Abstract: Double Field Theory in a nutshell
• Double Field Theory (DFT) formally uses doubled, (D +
D)-dimensonal coordinates, tomanifest O(D,D) symmetry and to unify
diffeomorphisms and B-field gauge symmetry.
• It looks like a (D + D)-dimensonal theory, but the theory is
required to satisfy so-called‘section condition’ such that it lives
on a D-dimensional hyperspace, i.e. ‘section’.
• DFT assumes the whole massless NS-NS sector as the
gravitational unity.– The underlying differential geometry is
genuinely ‘stringy’ beyond Riemann.
• DFT is formulated in terms of its own field variables, VAp,
V̄Ap̄, d , which are strictlyO(D,D) covariant. The connection to GR
can be only established after parametrizingthem by the conventional
Riemannian variables, such as gµν , eµa, Bµν , φ.
• Covariant derivatives and curvatures have been constructed,
and successfully applied– to identify the expression, R + 4 |dφ|2 −
112 |dB|
2, as a ‘scalar curvature’ of DFT;– to unify IIA and IIB SUGRAs
into D = 10 maximally supersymmetric DFT;– to couple D = 4 DFT to
the Standard Model unambiguously.
• Each term in every formula is manifestly covariant for the
Fundamental Symmetries:∗ O(D,D) T-duality∗ DFT-diffeomorphisms
(diffeomorphisms plus B-field gauge symmetry)∗ A pair of local
Lorentz symmetries, Spin(1,D−1)L × Spin(D−1, 1)R∗ ‘Coordinate gauge
symmetry’ (section condition)
# The self-interaction of the NS-NS sector modifies GR at
‘short’ distance (R/MG ), andmay solve the DM problem in ‘uroboros’
manner.
# Superstring theory itself is better formulated in terms of
doubled geometry.JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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Talk Organization
I. Geometric formulation of DFT & Coupling to the Standard
Model
II. Doubled-yet-gauged coordinates
III. ‘Uroboros’ solution to the Dark Matter Problem via DFT
∗ Based on works in collaborations with Imtak Jeon, Kanghoon
Lee, Yoonji Suh,Wonyoung Cho, Jose Fernández-Melgarejo, Soo-Jong
Rey, Woohyun Rim,Yuho Sakatani, Sung Moon Ko, Minwoo Suh, Kang-Sin
Choi, Rene Meyér,Charles Melby-Thompson, Chris Blair, Emanuel
Malek, and Xavier Bekaert.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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I. Geometric Formulation
• Stringy differential geometry, beyond RiemannImtak Jeon,
Kanghoon Lee, JHP 1105.6294
• Stringy Unification of IIA and IIB Supergravities under N= 2
D= 10 SupersymmetricDouble Field Theory Imtak Jeon, Kanghoon Lee,
JHP, Yoonji Suh 1210.5078
•Supersymmetric gauged Double Field Theory: Systematic
derivation by virtue of TwistWonyoung Cho, Jose J.
Fernández-Melgarejo, Imtak Jeon, JHP 1505.01301
• Standard Model as a Double Field Theory Kang-Sin Choi, JHP
1506.05277
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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• Notation
Index Representation Metric (raising/lowering indices)
A,B, · · · O(D,D) & Diffeomorphism vector JAB =
0 1
1 0
p, q, · · · Spin(1,D−1)L vector ηpq = diag(−+ + · · ·+)
α, β, · · · Spin(1,D−1)L spinor Cαβ , (γp)T = CγpC−1
p̄, q̄, · · · Spin(D−1, 1)R vector η̄p̄q̄ = diag(+−− · · ·−)
ᾱ, β̄, · · · Spin(D−1, 1)R spinor C̄ᾱβ̄ , (γ̄p̄)T =
C̄γ̄p̄C̄−1
– Here D denotes the dimenison of the physical spacetime. In
this talk, D ≡ 4 or 10.
– The constant O(D,D) metric, JAB , naturally decomposes the
doubled coordinatesof DFT into two parts,
xA = (x̃µ, xν) , ∂A = (∂̃µ, ∂ν) ,
where µ, ν are D-diemensional curved indices.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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• Fundamental fields of DFT
VAp , V̄Ap̄ , d : Geometric and hence Gravitational
These represent the massless NS-NS sector in string theory, c.f.
R-R sector, Cαᾱ .
– The pair of vielbeins satisfy four defining properties,
VApV Aq = ηpq , V̄Ap̄V̄ Aq̄ = η̄p̄q̄ VApV̄ Aq̄ = 0 , VApVBp +
V̄Ap̄V̄Bp̄ = JAB ,
such that they are the “square-roots" of projectors,
PAB = VApV Bp , P̄AB = V̄Ap̄V̄ Bp̄
satisfying
P2 = P , P̄2 = P̄ , PP̄ = 0 , P + P̄ = 1 .
– The dilaton gives rise to the O(D,D) invariant integral
measure with weight one,after exponentiation:
e−2d
Naturally the cosmological constant term in DFT should be given
by e−2d ΛDFTwhich differs from the conventional one in Riemannian
GR, and hencereformulates the ‘cosmological constant problem’.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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• Fundamental fields of DFT
VAp , V̄Ap̄ , d : Geometric and hence Gravitational
These represent the massless NS-NS sector in string theory, c.f.
R-R sector, Cαᾱ .
– The pair of vielbeins satisfy four defining properties,
VApV Aq = ηpq , V̄Ap̄V̄ Aq̄ = η̄p̄q̄ VApV̄ Aq̄ = 0 , VApVBp +
V̄Ap̄V̄Bp̄ = JAB ,
such that they are the “square-roots" of projectors,
PAB = VApV Bp , P̄AB = V̄Ap̄V̄ Bp̄
satisfying
P2 = P , P̄2 = P̄ , PP̄ = 0 , P + P̄ = 1 .
– The dilaton gives rise to the O(D,D) invariant integral
measure with weight one,after exponentiation:
e−2d
Naturally the cosmological constant term in DFT should be given
by e−2d ΛDFTwhich differs from the conventional one in Riemannian
GR, and hencereformulates the ‘cosmological constant problem’.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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• Fundamental fields of DFT
VAp , V̄Ap̄ , d : Geometric and hence Gravitational
These represent the massless NS-NS sector in string theory, c.f.
R-R sector, Cαᾱ .
– The pair of vielbeins satisfy four defining properties,
VApV Aq = ηpq , V̄Ap̄V̄ Aq̄ = η̄p̄q̄ VApV̄ Aq̄ = 0 , VApVBp +
V̄Ap̄V̄Bp̄ = JAB ,
such that they are the “square-roots" of projectors,
PAB = VApV Bp , P̄AB = V̄Ap̄V̄ Bp̄
satisfying
P2 = P , P̄2 = P̄ , PP̄ = 0 , P + P̄ = 1 .
– The dilaton gives rise to the O(D,D) invariant integral
measure with weight one,after exponentiation:
e−2d
Naturally the cosmological constant term in DFT should be given
by e−2d ΛDFTwhich differs from the conventional one in Riemannian
GR, and hencereformulates the ‘cosmological constant problem’.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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• Semi-covariant derivative : Jeon-Lee-JHP 2010, 2011
∇CTA1A2···An := ∂CTA1A2···An − ωT ΓB
BCTA1A2···An +n∑
i=1
ΓCAiBTA1···Ai−1BAi+1···An .
The DFT-version of the Christoffel connection has been uniquely
determined,
ΓCAB=2(P∂C PP̄)[AB]+2(
P̄[AD P̄B]
E−P[ADPB]
E)∂DPEC− 4D−1
(P̄C[AP̄B]
D+PC[APB]D)(∂Dd+(P∂
E PP̄)[ED])
by demanding the compatibility with the NS-NS sector, ∇APBC =
∇AP̄BC = ∇Ad = 0,plus some extra ‘torsionless’ conditions.
• Semi-covariant Riemann-like curvature :
SABCD := 12(
RABCD + RCDAB − ΓE ABΓECD)
where RABCD denotes the ordinary “field strength” of a
connection,
RCDAB=∂AΓBCD−∂BΓACD+ΓACE ΓBED−ΓBC
E ΓAED ⇐ dΓ+Γ∧Γ .
Under arbitrary transformation of the connection, it transforms
as ‘total derivative’,
δSABCD = ∇[AδΓB]CD +∇[CδΓD]AB ,
and further satisfies
SABCD=S[AB][CD]=SCDAB , S[ABC]D=0 , PIAPJ
B P̄KC P̄L
DSABCD=0 , PIAP̄J
BPKC P̄L
DSABCD=0 .
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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• Semi-covariant ‘Master’ derivative :DA := ∂A + ΓA + ΦA + Φ̄A =
∇A + ΦA + Φ̄A .
where the spin connections are determined in terms of the
Christoffel-like connectionby requiring the compatibility with the
vielbeins,
DAVBp = ∇AVBp + ΦApqVBq = 0 , DAV̄Bp̄ = ∇AV̄Bp̄ + Φ̄Ap̄q̄V̄Bq̄ =
0 .
• Complete covariatizations : 〈 divergences, Laplacians, Dirac
operators and curvatures 〉
PC DP̄A1B1 · · · P̄An Bn∇DTB1···Bn =⇒ DpTq̄1q̄2···q̄n ,
P̄C DPA1B1 · · ·PAn Bn∇DTB1···Bn =⇒ Dp̄Tq1q2···qn ,
DpTpq̄1q̄2···q̄n , Dp̄Tp̄q1q2···qn , DpD
pTq̄1q̄2···q̄n , Dp̄Dp̄Tq1q2···qn ,
γpDpρ , γ̄p̄Dp̄ρ′ , Dp̄ρ , Dpρ′ , γpDpψq̄ , γ̄p̄Dp̄ψ′q , Dp̄ψp̄
, Dpψ′p ,
D±C := γpDpC ± γ(D+1)Dp̄Cγ̄p̄ , (D±)2 = 0 =⇒ F := D+C (RR field
strength ) ,
PAC P̄BDSCD (Ricci-like ) , (PACPBD − P̄AC P̄BD)SABCD ( scalar )
.
# Combining the curvatures, we also have the ‘conserved’
Einstein-like curvature:
∇AGAB = 0 , GAB := 2(PAC P̄BD − P̄ACPBD)SCD − 12JAB(Spqpq −
Sp̄q̄ p̄q̄) .
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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• Semi-covariant ‘Master’ derivative :DA := ∂A + ΓA + ΦA + Φ̄A =
∇A + ΦA + Φ̄A .
where the spin connections are determined in terms of the
Christoffel-like connectionby requiring the compatibility with the
vielbeins,
DAVBp = ∇AVBp + ΦApqVBq = 0 , DAV̄Bp̄ = ∇AV̄Bp̄ + Φ̄Ap̄q̄V̄Bq̄ =
0 .
• Complete covariatizations : 〈 divergences, Laplacians, Dirac
operators and curvatures 〉
PC DP̄A1B1 · · · P̄An Bn∇DTB1···Bn =⇒ DpTq̄1q̄2···q̄n ,
P̄C DPA1B1 · · ·PAn Bn∇DTB1···Bn =⇒ Dp̄Tq1q2···qn ,
DpTpq̄1q̄2···q̄n , Dp̄Tp̄q1q2···qn , DpD
pTq̄1q̄2···q̄n , Dp̄Dp̄Tq1q2···qn ,
γpDpρ , γ̄p̄Dp̄ρ′ , Dp̄ρ , Dpρ′ , γpDpψq̄ , γ̄p̄Dp̄ψ′q , Dp̄ψp̄
, Dpψ′p ,
D±C := γpDpC ± γ(D+1)Dp̄Cγ̄p̄ , (D±)2 = 0 =⇒ F := D+C (RR field
strength ) ,
PAC P̄BDSCD (Ricci-like ) , (PACPBD − P̄AC P̄BD)SABCD ( scalar )
.
# Combining the curvatures, we also have the ‘conserved’
Einstein-like curvature:
∇AGAB = 0 , GAB := 2(PAC P̄BD − P̄ACPBD)SCD − 12JAB(Spqpq −
Sp̄q̄ p̄q̄) .
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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• Type II D = 10 Maximally Supersymmetric Double Field Theory :
Jeon-Lee-JHP-Suh 2012
LMax = e−2d[
18 (P
ABPCD − P̄ABP̄CD)SACBD + 12Tr(FF̄) + i ρ̄Fρ′ + iψ̄p̄γqF
γ̄p̄ψ′q
+i 12 ρ̄γpDpρ− iψ̄p̄Dp̄ρ− i 12 ψ̄
p̄γqDqψp̄ − i 12 ρ̄′γ̄p̄Dp̄ρ′ + iψ̄′pDpρ′ + i 12 ψ̄
′p γ̄q̄Dq̄ψ′p]
– Due to the twofold spin groups, Spin(1, 9)L × Spin(9, 1)R ,
the theory unifies theconventional IIA and IIB SUGRAs. Namely the
theory is chiral w.r.t. both spingroups and hence unique. IIA and
IIB appear as two distinct types of solutions.
– Maximal 16 + 16 local SUSY (full order construction realizing
‘1.5 formalism’).
– Euler-Lagrange equations include the DFT version of the
Einstein equation:
Spq̄︸︷︷︸curvature
= −Tr(γpF γ̄q̄F̄) + fermions︸ ︷︷ ︸matters
,
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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• Type II D = 10 Maximally Supersymmetric Double Field Theory :
Jeon-Lee-JHP-Suh 2012
LMax = e−2d[
18 (P
ABPCD − P̄ABP̄CD)SACBD + 12Tr(FF̄) + i ρ̄Fρ′ + iψ̄p̄γqF
γ̄p̄ψ′q
+i 12 ρ̄γpDpρ− iψ̄p̄Dp̄ρ− i 12 ψ̄
p̄γqDqψp̄ − i 12 ρ̄′γ̄p̄Dp̄ρ′ + iψ̄′pDpρ′ + i 12 ψ̄
′p γ̄q̄Dq̄ψ′p]
– Due to the twofold spin groups, Spin(1, 9)L × Spin(9, 1)R ,
the theory unifies theconventional IIA and IIB SUGRAs. Namely the
theory is chiral w.r.t. both spingroups and hence unique. IIA and
IIB appear as two distinct types of solutions.
– Maximal 16 + 16 local SUSY (full order construction realizing
‘1.5 formalism’).
– Euler-Lagrange equations include the DFT version of the
Einstein equation:
Spq̄︸︷︷︸curvature
= −Tr(γpF γ̄q̄F̄) + fermions︸ ︷︷ ︸matters
,
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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• Type II D = 10 Maximally Supersymmetric Double Field Theory :
Jeon-Lee-JHP-Suh 2012
LMax = e−2d[
18 (P
ABPCD − P̄ABP̄CD)SACBD + 12Tr(FF̄) + i ρ̄Fρ′ + iψ̄p̄γqF
γ̄p̄ψ′q
+i 12 ρ̄γpDpρ− iψ̄p̄Dp̄ρ− i 12 ψ̄
p̄γqDqψp̄ − i 12 ρ̄′γ̄p̄Dp̄ρ′ + iψ̄′pDpρ′ + i 12 ψ̄
′p γ̄q̄Dq̄ψ′p]
– Due to the twofold spin groups, Spin(1, 9)L × Spin(9, 1)R ,
the theory unifies theconventional IIA and IIB SUGRAs. Namely the
theory is chiral w.r.t. both spingroups and hence unique. IIA and
IIB appear as two distinct types of solutions.
– Maximal 16 + 16 local SUSY (full order construction realizing
‘1.5 formalism’).
– Euler-Lagrange equations include the DFT version of the
Einstein equation:
Spq̄︸︷︷︸curvature
= −Tr(γpF γ̄q̄F̄) + fermions︸ ︷︷ ︸matters
,
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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• Yang-Mills: Jeon-Lee-JHP 2011
– Completely covariant Yang-Mills field strength is given by
PAM P̄BNFMN
where FMN is the semi-covariant field strength of a YM
potential, VM ,
FMN := ∇MVN −∇NVM − i [VM ,VN ] .
– It is fully covariant w.r.t. all the DFT symmetries plus YM
gauge symmetry.
– We can freely impose O(D,D) & YM gauge covariant
conditions on the potential:
VMVM = 0 , VM∂M = 0 ,
in order not to double the physical degrees.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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• Yang-Mills: Jeon-Lee-JHP 2011
– Completely covariant Yang-Mills field strength is given by
PAM P̄BNFMN
where FMN is the semi-covariant field strength of a YM
potential, VM ,
FMN := ∇MVN −∇NVM − i [VM ,VN ] .
– It is fully covariant w.r.t. all the DFT symmetries plus YM
gauge symmetry.
– We can freely impose O(D,D) & YM gauge covariant
conditions on the potential:
VMVM = 0 , VM∂M = 0 ,
in order not to double the physical degrees.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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• Yang-Mills: Jeon-Lee-JHP 2011
– Completely covariant Yang-Mills field strength is given by
PAM P̄BNFMN
where FMN is the semi-covariant field strength of a YM
potential, VM ,
FMN := ∇MVN −∇NVM − i [VM ,VN ] .
– It is fully covariant w.r.t. all the DFT symmetries plus YM
gauge symmetry.
– We can freely impose O(D,D) & YM gauge covariant
conditions on the potential:
VMVM = 0 , VM∂M = 0 ,
in order not to double the physical degrees.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
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Coupling to the Standard Model
• D = 4 DFT naturally, or minimally, couples to the Standard
Model, in a completelycovariant and unambiguous manner:
– O(4, 4) T-duality– Twofold local Lorentz symmetry, Spin(1, 3)L
× Spin(3, 1)R– DFT-diffeomorphisms– SU(3)× SU(2)×U(1) gauge
symmetry
LSM−DFT = e−2d
1
16πGN(PABPCD − P̄ABP̄CD)SACBD
+∑A P
ABP̄CDTr(FACFBD) +∑ψ ψ̄γ
aDaψ +∑ψ′ ψ̄
′γ̄āDāψ′
−HAB(DAφ)†DBφ − V (φ) + yd q̄·φ d + yu q̄·φ̃ u + ye l̄ ′·φ
e′
which reduces to the ‘standard’ SM on trivial flat background
after gauge fixings.
Choi-JHP 2015 [PRL]
• While coupling DFT to SM, one has to decide the spin group for
each fermion:It is a prediction of DFT that the spin group is
twofold: Spin(1, 3)L vs. Spin(3, 1)R .
• No experimental evidence of proton decay lead us to
‘conjecture’ that the quarks andthe leptons may belong to the
distinct spin groups, which forbids a class of higherorder terms:
e.g. a bi-quark vector and a bi-lepton vector cannot be
contracted.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Coupling to the Standard Model
• D = 4 DFT naturally, or minimally, couples to the Standard
Model, in a completelycovariant and unambiguous manner:
– O(4, 4) T-duality– Twofold local Lorentz symmetry, Spin(1, 3)L
× Spin(3, 1)R– DFT-diffeomorphisms– SU(3)× SU(2)×U(1) gauge
symmetry
LSM−DFT = e−2d
1
16πGN(PABPCD − P̄ABP̄CD)SACBD
+∑A P
ABP̄CDTr(FACFBD) +∑ψ ψ̄γ
aDaψ +∑ψ′ ψ̄
′γ̄āDāψ′
−HAB(DAφ)†DBφ − V (φ) + yd q̄·φ d + yu q̄·φ̃ u + ye l̄ ′·φ
e′
which reduces to the ‘standard’ SM on trivial flat background
after gauge fixings.
Choi-JHP 2015 [PRL]
• While coupling DFT to SM, one has to decide the spin group for
each fermion:It is a prediction of DFT that the spin group is
twofold: Spin(1, 3)L vs. Spin(3, 1)R .
• No experimental evidence of proton decay lead us to
‘conjecture’ that the quarks andthe leptons may belong to the
distinct spin groups, which forbids a class of higherorder terms:
e.g. a bi-quark vector and a bi-lepton vector cannot be
contracted.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Coupling to the Standard Model
• D = 4 DFT naturally, or minimally, couples to the Standard
Model, in a completelycovariant and unambiguous manner:
– O(4, 4) T-duality– Twofold local Lorentz symmetry, Spin(1, 3)L
× Spin(3, 1)R– DFT-diffeomorphisms– SU(3)× SU(2)×U(1) gauge
symmetry
LSM−DFT = e−2d
1
16πGN(PABPCD − P̄ABP̄CD)SACBD
+∑A P
ABP̄CDTr(FACFBD) +∑ψ ψ̄γ
aDaψ +∑ψ′ ψ̄
′γ̄āDāψ′
−HAB(DAφ)†DBφ − V (φ) + yd q̄·φ d + yu q̄·φ̃ u + ye l̄ ′·φ
e′
which reduces to the ‘standard’ SM on trivial flat background
after gauge fixings.
Choi-JHP 2015 [PRL]
• While coupling DFT to SM, one has to decide the spin group for
each fermion:It is a prediction of DFT that the spin group is
twofold: Spin(1, 3)L vs. Spin(3, 1)R .
• No experimental evidence of proton decay lead us to
‘conjecture’ that the quarks andthe leptons may belong to the
distinct spin groups, which forbids a class of higherorder terms:
e.g. a bi-quark vector and a bi-lepton vector cannot be
contracted.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• In the above constructions of DFTs, there is something I have
not yet covered:
– The doubled coordinates, xA, A = 1, 2, · · · ,D + D, and the
associated doubledderivatives, ∂A, need to be ‘halved’.
– It is done in DFT by imposing an O(D,D) covariant constraint,
so-called the‘section condition’,
∂A∂A anything = 0 .
– Explicitly, for arbitrary functions, Φ, Φ̂, the section
condition means
∂A∂AΦ=0 , ∂A∂A
(ΦΦ̂)
=0 =⇒ ∂AΦ∂AΦ̂=0 .
– With the O(D,D) metric, JAB =(
0 1
1 0
), and the doubled coordinates,
xA = (x̃µ, xν), ∂A = (∂̃µ, ∂ν), we get
∂A∂A = 2∂µ∂̃µ .
– The section condition can be then conveniently solved by
setting ∂̃µ ≡ 0.The most general solutions are then generated by
its O(D,D) rotations.
# DFT lives on a D-dimensional hyperspace, i.e. section.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• In the above constructions of DFTs, there is something I have
not yet covered:
– The doubled coordinates, xA, A = 1, 2, · · · ,D + D, and the
associated doubledderivatives, ∂A, need to be ‘halved’.
– It is done in DFT by imposing an O(D,D) covariant constraint,
so-called the‘section condition’,
∂A∂A anything = 0 .
– Explicitly, for arbitrary functions, Φ, Φ̂, the section
condition means
∂A∂AΦ=0 , ∂A∂A
(ΦΦ̂)
=0 =⇒ ∂AΦ∂AΦ̂=0 .
– With the O(D,D) metric, JAB =(
0 1
1 0
), and the doubled coordinates,
xA = (x̃µ, xν), ∂A = (∂̃µ, ∂ν), we get
∂A∂A = 2∂µ∂̃µ .
– The section condition can be then conveniently solved by
setting ∂̃µ ≡ 0.The most general solutions are then generated by
its O(D,D) rotations.
# DFT lives on a D-dimensional hyperspace, i.e. section.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• In the above constructions of DFTs, there is something I have
not yet covered:
– The doubled coordinates, xA, A = 1, 2, · · · ,D + D, and the
associated doubledderivatives, ∂A, need to be ‘halved’.
– It is done in DFT by imposing an O(D,D) covariant constraint,
so-called the‘section condition’,
∂A∂A anything = 0 .
– Explicitly, for arbitrary functions, Φ, Φ̂, the section
condition means
∂A∂AΦ=0 , ∂A∂A
(ΦΦ̂)
=0 =⇒ ∂AΦ∂AΦ̂=0 .
– With the O(D,D) metric, JAB =(
0 1
1 0
), and the doubled coordinates,
xA = (x̃µ, xν), ∂A = (∂̃µ, ∂ν), we get
∂A∂A = 2∂µ∂̃µ .
– The section condition can be then conveniently solved by
setting ∂̃µ ≡ 0.The most general solutions are then generated by
its O(D,D) rotations.
# DFT lives on a D-dimensional hyperspace, i.e. section.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
DFT backgrounds : Riemannian IIA/IIB vs. non-Riemannian IIC
• W.r.t. ∂∂x̃µ≡ 0, the DFT-vielbeins and the DFT-dilaton can be
generically solved and
parametrized by a pair of ordinary vierbeins, eµp, ēµp̄ and a
B-field:
VMp ≡ 1√2
(e−1)pµ(B + e)νp
, V̄Mp̄ ≡ 1√2 (ē−1)p̄µ
(B + ē)νp̄
, e−2d ≡√|g|e−2φ ,where the two vierbeins must correspond to the
same Riemannian metric,
eµpeνqηpq = −ēµ p̄ ēν q̄ η̄p̄q̄ ≡ gµν .Jeon-Lee-JHP-Suh
2012
– It follows that (e−1ē)pp̄ is a Lorentz rotation, and
hence,
det(e−1ē) = +1 : type IIA vs. det(e−1ē) = −1 : type IIB
– DFT-metric (“generalized metric” a la Siegel, Hull, Zwiebach)
reads then
HMN := PAB − P̄AB = VApVBp − V̄Ap̄V̄Bp̄ ≡
g−1 −g−1BBg−1 g − Bg−1B
.• The above is not the most general parametrization: there
exists a class of DFT
backgrounds which do not admit any Riemannian interpretation ⇒
type IIC JHP 2016– Such non-Riemannian brackgrounds lead to chiral
or non-relativistic string theory
a la Gomis-Ooguri. Ko-MelbyThompson-Meyer-JHP 2015
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
DFT backgrounds : Riemannian IIA/IIB vs. non-Riemannian IIC
• W.r.t. ∂∂x̃µ≡ 0, the DFT-vielbeins and the DFT-dilaton can be
generically solved and
parametrized by a pair of ordinary vierbeins, eµp, ēµp̄ and a
B-field:
VMp ≡ 1√2
(e−1)pµ(B + e)νp
, V̄Mp̄ ≡ 1√2 (ē−1)p̄µ
(B + ē)νp̄
, e−2d ≡√|g|e−2φ ,where the two vierbeins must correspond to the
same Riemannian metric,
eµpeνqηpq = −ēµ p̄ ēν q̄ η̄p̄q̄ ≡ gµν .Jeon-Lee-JHP-Suh
2012
– It follows that (e−1ē)pp̄ is a Lorentz rotation, and
hence,
det(e−1ē) = +1 : type IIA vs. det(e−1ē) = −1 : type IIB
– DFT-metric (“generalized metric” a la Siegel, Hull, Zwiebach)
reads then
HMN := PAB − P̄AB = VApVBp − V̄Ap̄V̄Bp̄ ≡
g−1 −g−1BBg−1 g − Bg−1B
.• The above is not the most general parametrization: there
exists a class of DFT
backgrounds which do not admit any Riemannian interpretation ⇒
type IIC JHP 2016– Such non-Riemannian brackgrounds lead to chiral
or non-relativistic string theory
a la Gomis-Ooguri. Ko-MelbyThompson-Meyer-JHP 2015
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
DFT backgrounds : Riemannian IIA/IIB vs. non-Riemannian IIC
• W.r.t. ∂∂x̃µ≡ 0, the DFT-vielbeins and the DFT-dilaton can be
generically solved and
parametrized by a pair of ordinary vierbeins, eµp, ēµp̄ and a
B-field:
VMp ≡ 1√2
(e−1)pµ(B + e)νp
, V̄Mp̄ ≡ 1√2 (ē−1)p̄µ
(B + ē)νp̄
, e−2d ≡√|g|e−2φ ,where the two vierbeins must correspond to the
same Riemannian metric,
eµpeνqηpq = −ēµ p̄ ēν q̄ η̄p̄q̄ ≡ gµν .Jeon-Lee-JHP-Suh
2012
– It follows that (e−1ē)pp̄ is a Lorentz rotation, and
hence,
det(e−1ē) = +1 : type IIA vs. det(e−1ē) = −1 : type IIB
– DFT-metric (“generalized metric” a la Siegel, Hull, Zwiebach)
reads then
HMN := PAB − P̄AB = VApVBp − V̄Ap̄V̄Bp̄ ≡
g−1 −g−1BBg−1 g − Bg−1B
.• The above is not the most general parametrization: there
exists a class of DFT
backgrounds which do not admit any Riemannian interpretation ⇒
type IIC JHP 2016– Such non-Riemannian brackgrounds lead to chiral
or non-relativistic string theory
a la Gomis-Ooguri. Ko-MelbyThompson-Meyer-JHP 2015
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
II. Doubled-yet-gauged coordinates
• Comments on double field theory and diffeomorphisms JHP
1304.5946
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• The section condition can be easily shown to be equivalent to
a particular type oftranslational invariance, in a self-consistent
way:
∂A∂A ≡ 0 ⇐⇒
[Φ̂(x + ∆) = Φ̂(x) , ∆A = Φ̃∂AΦ
]where Φ̂, Φ̃, Φ denote arbitrary functions in DFT, such that ∆A
= Φ̃∂AΦ generates themost general form of a
‘derivative-index-valued’ vector, to satisfy ∆A∂A = 0.
• This equivalence suggests that the doubled coordinates in DFT
are actually gauged:the doubled coordinate space is equipped with
an ‘equivalence relation’, JHP 2013
xA ∼ xA + ∆A where ∆A∂A = 0 .
which we call ‘Coordinate Gauge Symmetry’.
# For example, w.r.t. ∂∂x̃µ≡ 0, we have explicitly (x̃µ , xν)
∼
(x̃µ + Φ̃∂µΦ , xν
).
• Doubled-yet-gauged coordinates
Each equivalence class, or gauge orbit in RD+D ,represents a
single physical point in RD .
The claim is that, spacetime physics can bebetter understood in
terms of the
doubled-yet-gauged coordinate system.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• The section condition can be easily shown to be equivalent to
a particular type oftranslational invariance, in a self-consistent
way:
∂A∂A ≡ 0 ⇐⇒
[Φ̂(x + ∆) = Φ̂(x) , ∆A = Φ̃∂AΦ
]where Φ̂, Φ̃, Φ denote arbitrary functions in DFT, such that ∆A
= Φ̃∂AΦ generates themost general form of a
‘derivative-index-valued’ vector, to satisfy ∆A∂A = 0.
• This equivalence suggests that the doubled coordinates in DFT
are actually gauged:the doubled coordinate space is equipped with
an ‘equivalence relation’, JHP 2013
xA ∼ xA + ∆A where ∆A∂A = 0 .
which we call ‘Coordinate Gauge Symmetry’.
# For example, w.r.t. ∂∂x̃µ≡ 0, we have explicitly (x̃µ , xν)
∼
(x̃µ + Φ̃∂µΦ , xν
).
• Doubled-yet-gauged coordinates
Each equivalence class, or gauge orbit in RD+D ,represents a
single physical point in RD .
The claim is that, spacetime physics can bebetter understood in
terms of the
doubled-yet-gauged coordinate system.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• The section condition can be easily shown to be equivalent to
a particular type oftranslational invariance, in a self-consistent
way:
∂A∂A ≡ 0 ⇐⇒
[Φ̂(x + ∆) = Φ̂(x) , ∆A = Φ̃∂AΦ
]where Φ̂, Φ̃, Φ denote arbitrary functions in DFT, such that ∆A
= Φ̃∂AΦ generates themost general form of a
‘derivative-index-valued’ vector, to satisfy ∆A∂A = 0.
• This equivalence suggests that the doubled coordinates in DFT
are actually gauged:the doubled coordinate space is equipped with
an ‘equivalence relation’, JHP 2013
xA ∼ xA + ∆A where ∆A∂A = 0 .
which we call ‘Coordinate Gauge Symmetry’.
# For example, w.r.t. ∂∂x̃µ≡ 0, we have explicitly (x̃µ , xν)
∼
(x̃µ + Φ̃∂µΦ , xν
).
• Doubled-yet-gauged coordinates
Each equivalence class, or gauge orbit in RD+D ,represents a
single physical point in RD .
The claim is that, spacetime physics can bebetter understood in
terms of the
doubled-yet-gauged coordinate system.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• The section condition can be easily shown to be equivalent to
a particular type oftranslational invariance, in a self-consistent
way:
∂A∂A ≡ 0 ⇐⇒
[Φ̂(x + ∆) = Φ̂(x) , ∆A = Φ̃∂AΦ
]where Φ̂, Φ̃, Φ denote arbitrary functions in DFT, such that ∆A
= Φ̃∂AΦ generates themost general form of a
‘derivative-index-valued’ vector, to satisfy ∆A∂A = 0.
• This equivalence suggests that the doubled coordinates in DFT
are actually gauged:the doubled coordinate space is equipped with
an ‘equivalence relation’, JHP 2013
xA ∼ xA + ∆A where ∆A∂A = 0 .
which we call ‘Coordinate Gauge Symmetry’.
# For example, w.r.t. ∂∂x̃µ≡ 0, we have explicitly (x̃µ , xν)
∼
(x̃µ + Φ̃∂µΦ , xν
).
• Doubled-yet-gauged coordinates
Each equivalence class, or gauge orbit in RD+D ,represents a
single physical point in RD .
The claim is that, spacetime physics can bebetter understood in
terms of the
doubled-yet-gauged coordinate system.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• Diffeomorphisms in doubled-yet-gauged spacetime.
– Diffeomorphisms in doubled-yet-gauged spacetime are generated
by a generalizedLie derivative,
L̂VTM1···Mn := VN∂NTM1···Mn +ωT ∂NV
NTM1···Mn +n∑
i=1
(∂MiVN−∂NVMi )TM1···Mi−1N
Mi+1···Mn
where ωT denotes the weight. Siegel, c.f. Courant
– In particular, the generalized Lie derivative of the O(D,D)
invariant metricvanishes
L̂VJAB = 0 .
– The commutator is closed by C-bracket,[L̂U , L̂V
]= L̂[U,V]C , [U ,V]
MC := U
N∂NVM − VN∂NUM + 12VN∂MUN − 12U
N∂MVN
Hull-Zwiebach
# DFT-diffeomorphisms decompose into undoubled Riemannian
diffeomorphismsand B-field gauge symmetry,
VM = (λµ , ξν) =⇒ δBµν = ∂µλν − ∂νλµ , δxµ = ξµ .
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• Diffeomorphisms in doubled-yet-gauged spacetime.
– Diffeomorphisms in doubled-yet-gauged spacetime are generated
by a generalizedLie derivative,
L̂VTM1···Mn := VN∂NTM1···Mn +ωT ∂NV
NTM1···Mn +n∑
i=1
(∂MiVN−∂NVMi )TM1···Mi−1N
Mi+1···Mn
where ωT denotes the weight. Siegel, c.f. Courant
– In particular, the generalized Lie derivative of the O(D,D)
invariant metricvanishes
L̂VJAB = 0 .
– The commutator is closed by C-bracket,[L̂U , L̂V
]= L̂[U,V]C , [U ,V]
MC := U
N∂NVM − VN∂NUM + 12VN∂MUN − 12U
N∂MVN
Hull-Zwiebach
# DFT-diffeomorphisms decompose into undoubled Riemannian
diffeomorphismsand B-field gauge symmetry,
VM = (λµ , ξν) =⇒ δBµν = ∂µλν − ∂νλµ , δxµ = ξµ .
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• Diffeomorphisms in doubled-yet-gauged spacetime.
– Diffeomorphisms in doubled-yet-gauged spacetime are generated
by a generalizedLie derivative,
L̂VTM1···Mn := VN∂NTM1···Mn +ωT ∂NV
NTM1···Mn +n∑
i=1
(∂MiVN−∂NVMi )TM1···Mi−1N
Mi+1···Mn
where ωT denotes the weight. Siegel, c.f. Courant
– In particular, the generalized Lie derivative of the O(D,D)
invariant metricvanishes
L̂VJAB = 0 .
– The commutator is closed by C-bracket,[L̂U , L̂V
]= L̂[U,V]C , [U ,V]
MC := U
N∂NVM − VN∂NUM + 12VN∂MUN − 12U
N∂MVN
Hull-Zwiebach
# DFT-diffeomorphisms decompose into undoubled Riemannian
diffeomorphismsand B-field gauge symmetry,
VM = (λµ , ξν) =⇒ δBµν = ∂µλν − ∂νλµ , δxµ = ξµ .
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Problems with the usual infinitesimal one-form, dxM
• In the doubled-yet-gauged spacetime, the usual infinitesimal
one-form, dxM , is neithercovariant under DFT-diffeomorphisms,
δxM = VM , δ(dxM ) = dxN∂NVM 6= (∂NVM − ∂MVN )dxN ,
nor invariant under coordinate gauge symmetry,
dxM −→ d(xM + Φ̃∂M Φ
)6= dxM .
=⇒ The naive contraction with the DFT-metric, dxMdxNHMN , is not
a scalar, andthus cannot be used to define a ‘proper length’ in
DFT.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Doubled-yet-gauged coordinates & Gauged infinitesimal
one-form, DxM
• These problems can be all cured by gauging the infinitesimal
one-form,
DxM := dxM −AM .
– The gauge potential should satisfy the same property as the
coordinate gaugesymmetry generator (derivative-index-valued vector,
∆M = Φ̃∂M Φ), such that
AM∂M = 0 , AMAM = 0 .
Essentially, half of the components are trivial, for example
w.r.t. ∂∂x̃µ≡ 0,
AM = Aλ∂M xλ = (Aµ , 0) , DxM = (dx̃µ − Aµ , dxν) .
• With the appropriate transformations of the gauge potential,
the coordinate gaugesymmetry invariance and the DFT-diffeomorphism
covariance of DxM can be assured:
δC.G. xM = Φ̃∂M Φ , δC.G.AM = d(
Φ̃∂M Φ), δC.G. (DxM ) = 0 ;
δxM = VM , δAM = −∂MVNAN + ∂MVNdxN , δ(DxM ) = (∂NVM − ∂MVN )DxN
.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Doubled-yet-gauged coordinates & Gauged infinitesimal
one-form, DxM
• These problems can be all cured by gauging the infinitesimal
one-form,
DxM := dxM −AM .
– The gauge potential should satisfy the same property as the
coordinate gaugesymmetry generator (derivative-index-valued vector,
∆M = Φ̃∂M Φ), such that
AM∂M = 0 , AMAM = 0 .
Essentially, half of the components are trivial, for example
w.r.t. ∂∂x̃µ≡ 0,
AM = Aλ∂M xλ = (Aµ , 0) , DxM = (dx̃µ − Aµ , dxν) .
• With the appropriate transformations of the gauge potential,
the coordinate gaugesymmetry invariance and the DFT-diffeomorphism
covariance of DxM can be assured:
δC.G. xM = Φ̃∂M Φ , δC.G.AM = d(
Φ̃∂M Φ), δC.G. (DxM ) = 0 ;
δxM = VM , δAM = −∂MVNAN + ∂MVNdxN , δ(DxM ) = (∂NVM − ∂MVN )DxN
.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Length in the doubled-yet-gauged spacetime
• The proper length is defined through a path integral,
Length := − ln[ ∫DA exp
(−∫ √
DxM DxNHMN)]
.
• For the Riemannian DFT-metric, we have
DxM DxNHMN ≡ dxµdxνgµν + (dx̃µ − Aµ + dxρBρµ) (dx̃ν − Aν +
dxσBσν) gµν ,
and hence, after integrating out the gauge potential, Aµ, the
above O(D,D) covariantpath integral definition of the length
reduces to the conventional one,
Length =⇒∫ √
dxµdxνgµν .
# Apparently, being x̃µ-independent, it measures the distance
between two gauge orbitsrather than two points in RD+D , which is
of course a desired feature.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Length in the doubled-yet-gauged spacetime
• The proper length is defined through a path integral,
Length := − ln[ ∫DA exp
(−∫ √
DxM DxNHMN)]
.
• For the Riemannian DFT-metric, we have
DxM DxNHMN ≡ dxµdxνgµν + (dx̃µ − Aµ + dxρBρµ) (dx̃ν − Aν +
dxσBσν) gµν ,
and hence, after integrating out the gauge potential, Aµ, the
above O(D,D) covariantpath integral definition of the length
reduces to the conventional one,
Length =⇒∫ √
dxµdxνgµν .
# Apparently, being x̃µ-independent, it measures the distance
between two gauge orbitsrather than two points in RD+D , which is
of course a desired feature.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Length in the doubled-yet-gauged spacetime
• The proper length is defined through a path integral,
Length := − ln[ ∫DA exp
(−∫ √
DxM DxNHMN)]
.
• For the Riemannian DFT-metric, we have
DxM DxNHMN ≡ dxµdxνgµν + (dx̃µ − Aµ + dxρBρµ) (dx̃ν − Aν +
dxσBσν) gµν ,
and hence, after integrating out the gauge potential, Aµ, the
above O(D,D) covariantpath integral definition of the length
reduces to the conventional one,
Length =⇒∫ √
dxµdxνgµν .
# Apparently, being x̃µ-independent, it measures the distance
between two gauge orbitsrather than two points in RD+D , which is
of course a desired feature.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Point particle propagating in the doubled-yet-gauged
spacetime
• Point particle action on doubled-yet-gauged spacetime coupled
to the NS-NS sector:
Sparticle =∫
dτ[
e−1 DτX M DτX NHMN (X)− 14 m2e],
Ko-JHP-Suh 2016where e is an einbein and m is the mass of the
particle.
• With Riemannian DFT-metric, after integrating out e and AM ,
the above actionreduces to the conventional one for a relativistic
point particle now coupled to thestring frame metric only:
Sparticle ≡∫
dτ −m√−ẊµẊνgµν .
• This implies that the particle follows the geodesic path
defined in the string frame.• This preferred choice of the frame,
i.e. String frame over Einstein frame, is due to the
fundamental symmetries of DFT: O(D,D) symmetry ,
DFT-diffeomorphisms and thecoordinate gauge symmetry
# Newton mechanics can be also formulated in the
doubled-yet-gauged Euclidean space,
LNewton = 12 m Dt XM Dt X N δMN − V (X) ,
where M,N = 1, 2, · · · , 6 and the potential, V (X), satisfies
the section condition.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Point particle propagating in the doubled-yet-gauged
spacetime
• Point particle action on doubled-yet-gauged spacetime coupled
to the NS-NS sector:
Sparticle =∫
dτ[
e−1 DτX M DτX NHMN (X)− 14 m2e],
Ko-JHP-Suh 2016where e is an einbein and m is the mass of the
particle.
• With Riemannian DFT-metric, after integrating out e and AM ,
the above actionreduces to the conventional one for a relativistic
point particle now coupled to thestring frame metric only:
Sparticle ≡∫
dτ −m√−ẊµẊνgµν .
• This implies that the particle follows the geodesic path
defined in the string frame.• This preferred choice of the frame,
i.e. String frame over Einstein frame, is due to the
fundamental symmetries of DFT: O(D,D) symmetry ,
DFT-diffeomorphisms and thecoordinate gauge symmetry
# Newton mechanics can be also formulated in the
doubled-yet-gauged Euclidean space,
LNewton = 12 m Dt XM Dt X N δMN − V (X) ,
where M,N = 1, 2, · · · , 6 and the potential, V (X), satisfies
the section condition.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Point particle propagating in the doubled-yet-gauged
spacetime
• Point particle action on doubled-yet-gauged spacetime coupled
to the NS-NS sector:
Sparticle =∫
dτ[
e−1 DτX M DτX NHMN (X)− 14 m2e],
Ko-JHP-Suh 2016where e is an einbein and m is the mass of the
particle.
• With Riemannian DFT-metric, after integrating out e and AM ,
the above actionreduces to the conventional one for a relativistic
point particle now coupled to thestring frame metric only:
Sparticle ≡∫
dτ −m√−ẊµẊνgµν .
• This implies that the particle follows the geodesic path
defined in the string frame.• This preferred choice of the frame,
i.e. String frame over Einstein frame, is due to the
fundamental symmetries of DFT: O(D,D) symmetry ,
DFT-diffeomorphisms and thecoordinate gauge symmetry
# Newton mechanics can be also formulated in the
doubled-yet-gauged Euclidean space,
LNewton = 12 m Dt XM Dt X N δMN − V (X) ,
where M,N = 1, 2, · · · , 6 and the potential, V (X), satisfies
the section condition.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Point particle propagating in the doubled-yet-gauged
spacetime
• Point particle action on doubled-yet-gauged spacetime coupled
to the NS-NS sector:
Sparticle =∫
dτ[
e−1 DτX M DτX NHMN (X)− 14 m2e],
Ko-JHP-Suh 2016where e is an einbein and m is the mass of the
particle.
• With Riemannian DFT-metric, after integrating out e and AM ,
the above actionreduces to the conventional one for a relativistic
point particle now coupled to thestring frame metric only:
Sparticle ≡∫
dτ −m√−ẊµẊνgµν .
• This implies that the particle follows the geodesic path
defined in the string frame.• This preferred choice of the frame,
i.e. String frame over Einstein frame, is due to the
fundamental symmetries of DFT: O(D,D) symmetry ,
DFT-diffeomorphisms and thecoordinate gauge symmetry
# Newton mechanics can be also formulated in the
doubled-yet-gauged Euclidean space,
LNewton = 12 m Dt XM Dt X N δMN − V (X) ,
where M,N = 1, 2, · · · , 6 and the potential, V (X), satisfies
the section condition.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
String theory itself is better formulated
on doubled-yet-gauged spacetime:
• Covariant action for a string in doubled-yet-gauged
spacetimeKanghoon Lee, JHP 1307.8377
• Green-Schwarz superstring on doubled-yet-gauged spacetime JHP
1609.04265
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
String probes the doubled-yet-gauged spacetime
• The doubled-yet-gagued string action is, with Di X M = ∂i X M
−AMi ,
14πα′
∫d2σ Lstring , Lstring = − 12
√−h hij Di X M Dj X NHMN (X)− �ij Di X MAjM .
JHP-Lee 2013 (c.f. Hull 2006)
• The action is fully symmetric for an arbitrary curved
DFT-metric, HMN (X),essentially due to the auxiliary coordinate
gauge potential, AMi ,
– worldsheet diffeomorphisms plus Weyl symmetry– O(D,D)
T-duality– target spacetime DFT-diffeomorphisms– the coordinate
gauge symmetry : X M ∼ X M + Φ̃∂M Φ
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• With the Riemannian DFT-metric, after integrating out AM , the
doubled-yet-gaugedstring action reduces to the conventional
one,
14πα′Lstring ≡
12πα′
[− 12√−hhij∂i Xµ∂j Xνgµν(X) + 12 �
ij∂i Xµ∂j XνBµν(X) + 12 �ij∂i X̃µ∂j Xµ
],
with the bonus of the topological term introduced by
Giveon-Rocek; Hull.
– The EOM of AMi implies self-duality in the full doubled
spacetime,
HM NDi X N + 1√−h �ij Dj X M = 0 ,
which relates Xµ and X̃µ.
– The EOM of X M is identified as the Stringy Geodesic
Equation:
1√−h∂i
(√−hHLM Di X M
)+ ΓLMN (P̄M ADi X A)(PN BDi X B) = 0 .
• On the other hand, upon non-Riemannian backbrounds, the
doubled-yet-gauged stringaction leads to chiral or non-Relativistic
string theory a la Gomis-Ooguri.
Lee-JHP 2013, Ko-Melby-Thompson-Meyer-JHP 2015
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• With the Riemannian DFT-metric, after integrating out AM , the
doubled-yet-gaugedstring action reduces to the conventional
one,
14πα′Lstring ≡
12πα′
[− 12√−hhij∂i Xµ∂j Xνgµν(X) + 12 �
ij∂i Xµ∂j XνBµν(X) + 12 �ij∂i X̃µ∂j Xµ
],
with the bonus of the topological term introduced by
Giveon-Rocek; Hull.
– The EOM of AMi implies self-duality in the full doubled
spacetime,
HM NDi X N + 1√−h �ij Dj X M = 0 ,
which relates Xµ and X̃µ.
– The EOM of X M is identified as the Stringy Geodesic
Equation:
1√−h∂i
(√−hHLM Di X M
)+ ΓLMN (P̄M ADi X A)(PN BDi X B) = 0 .
• On the other hand, upon non-Riemannian backbrounds, the
doubled-yet-gauged stringaction leads to chiral or non-Relativistic
string theory a la Gomis-Ooguri.
Lee-JHP 2013, Ko-Melby-Thompson-Meyer-JHP 2015
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• With the Riemannian DFT-metric, after integrating out AM , the
doubled-yet-gaugedstring action reduces to the conventional
one,
14πα′Lstring ≡
12πα′
[− 12√−hhij∂i Xµ∂j Xνgµν(X) + 12 �
ij∂i Xµ∂j XνBµν(X) + 12 �ij∂i X̃µ∂j Xµ
],
with the bonus of the topological term introduced by
Giveon-Rocek; Hull.
– The EOM of AMi implies self-duality in the full doubled
spacetime,
HM NDi X N + 1√−h �ij Dj X M = 0 ,
which relates Xµ and X̃µ.
– The EOM of X M is identified as the Stringy Geodesic
Equation:
1√−h∂i
(√−hHLM Di X M
)+ ΓLMN (P̄M ADi X A)(PN BDi X B) = 0 .
• On the other hand, upon non-Riemannian backbrounds, the
doubled-yet-gauged stringaction leads to chiral or non-Relativistic
string theory a la Gomis-Ooguri.
Lee-JHP 2013, Ko-Melby-Thompson-Meyer-JHP 2015
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Supersymmetric extension
• The doubled-yet-gagued Green-Schwarz superstring action is
Ssuperstring = 14πα′∫
d2σ Lsuperstring ,
Lsuperstring = − 12√−hhij ΠMi Π
Nj HMN − �
ij Di X M(AjM − iΣjM
).
JHP 1609.04265
• Here, with a pair of Majorana-Weyl spinors, θα for Spin(1, 9)L
and θ′ᾱ for Spin(9, 1)R ,we set
ΠMi := Di XM − iΣMi , Σ
Mi := θ̄γ
M∂iθ + θ̄′γ̄M∂iθ
′ .
• Symmetries:– worldsheet diffeomorphisms plus Weyl symmetry–
O(D,D) T-duality– target spacetime DFT-diffeomorphisms– coordinate
gauge symmetry : X M ∼ X M + Φ̃∂M Φ– twofold Lorentz symmetry,
Spin(1, 9)L × Spin(9, 1)R ⇒ Unification of IIA & IIB– Maximal
16+16 SUSY & kappa symmetries
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Supersymmetric extension
• The doubled-yet-gagued Green-Schwarz superstring action is
Ssuperstring = 14πα′∫
d2σ Lsuperstring ,
Lsuperstring = − 12√−hhij ΠMi Π
Nj HMN − �
ij Di X M(AjM − iΣjM
).
JHP 1609.04265
• Here, with a pair of Majorana-Weyl spinors, θα for Spin(1, 9)L
and θ′ᾱ for Spin(9, 1)R ,we set
ΠMi := Di XM − iΣMi , Σ
Mi := θ̄γ
M∂iθ + θ̄′γ̄M∂iθ
′ .
• Symmetries:– worldsheet diffeomorphisms plus Weyl symmetry–
O(D,D) T-duality– target spacetime DFT-diffeomorphisms– coordinate
gauge symmetry : X M ∼ X M + Φ̃∂M Φ– twofold Lorentz symmetry,
Spin(1, 9)L × Spin(9, 1)R ⇒ Unification of IIA & IIB– Maximal
16+16 SUSY & kappa symmetries
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
III. Solution to the Dark Matter Problem
• The rotation curve of a point particle in stringy gravity
Sungmoon Ko, JHP, Minwoo Suh 1606.09307
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• The master derivative, DA, naturally provides the minimal
coupling of the StadardModel to DFT, or the massless NS-NS
sector.
“Symmetry dictates interaction”, C. N. Yang
i) Each SM fermion couples to the massless NS-NS sector as
e−2d ψ̄γADAψ = e−2d ψ̄γA(∂Aψ + 14 ΦApqγpqψ)
≡ 1√2
√−ge−2φ ψ̄γµ
(∂µψ +
14ωµpqγ
pqψ + 124 Hµpqγpqψ − ∂µφψ
)≡ 1√
2
√−g χ̄γµ
(∂µχ+
14ωµpqγ
pqχ+ 124 Hµpqγpqχ)
c.f. Coimbra-Strickland-Constable-Waldramwhere the field
redefinition of the fermion, χ ≡ e−φψ, has been performed
whichremoves the scalar dilaton completely. This result shows
that
– the scalar dilaton is transparent or ‘dark’ to the SM
fermions;– (not only F1 but also) the SM fermions can source the
H-flux!
ii) On the other hand, each SM gauge boson couples to the
massless NS-NS sector as
e−2d Tr(
PABP̄CDFACFBD)≡ − 14
√−ge−2φ Tr
(gκλgµνFκµFλν
)– B-field, or ‘axion’ (dual scalar), is dark to the gauge
bosons;– the Standard Model gauge bosons can source the scalar
dilaton, φ.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• The master derivative, DA, naturally provides the minimal
coupling of the StadardModel to DFT, or the massless NS-NS
sector.
“Symmetry dictates interaction”, C. N. Yang
i) Each SM fermion couples to the massless NS-NS sector as
e−2d ψ̄γADAψ = e−2d ψ̄γA(∂Aψ + 14 ΦApqγpqψ)
≡ 1√2
√−ge−2φ ψ̄γµ
(∂µψ +
14ωµpqγ
pqψ + 124 Hµpqγpqψ − ∂µφψ
)≡ 1√
2
√−g χ̄γµ
(∂µχ+
14ωµpqγ
pqχ+ 124 Hµpqγpqχ)
c.f. Coimbra-Strickland-Constable-Waldramwhere the field
redefinition of the fermion, χ ≡ e−φψ, has been performed
whichremoves the scalar dilaton completely. This result shows
that
– the scalar dilaton is transparent or ‘dark’ to the SM
fermions;– (not only F1 but also) the SM fermions can source the
H-flux!
ii) On the other hand, each SM gauge boson couples to the
massless NS-NS sector as
e−2d Tr(
PABP̄CDFACFBD)≡ − 14
√−ge−2φ Tr
(gκλgµνFκµFλν
)– B-field, or ‘axion’ (dual scalar), is dark to the gauge
bosons;– the Standard Model gauge bosons can source the scalar
dilaton, φ.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• The master derivative, DA, naturally provides the minimal
coupling of the StadardModel to DFT, or the massless NS-NS
sector.
“Symmetry dictates interaction”, C. N. Yang
i) Each SM fermion couples to the massless NS-NS sector as
e−2d ψ̄γADAψ = e−2d ψ̄γA(∂Aψ + 14 ΦApqγpqψ)
≡ 1√2
√−ge−2φ ψ̄γµ
(∂µψ +
14ωµpqγ
pqψ + 124 Hµpqγpqψ − ∂µφψ
)≡ 1√
2
√−g χ̄γµ
(∂µχ+
14ωµpqγ
pqχ+ 124 Hµpqγpqχ)
c.f. Coimbra-Strickland-Constable-Waldramwhere the field
redefinition of the fermion, χ ≡ e−φψ, has been performed
whichremoves the scalar dilaton completely. This result shows
that
– the scalar dilaton is transparent or ‘dark’ to the SM
fermions;– (not only F1 but also) the SM fermions can source the
H-flux!
ii) On the other hand, each SM gauge boson couples to the
massless NS-NS sector as
e−2d Tr(
PABP̄CDFACFBD)≡ − 14
√−ge−2φ Tr
(gκλgµνFκµFλν
)– B-field, or ‘axion’ (dual scalar), is dark to the gauge
bosons;– the Standard Model gauge bosons can source the scalar
dilaton, φ.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Spherical symmetry in DFT
• The coupling of DFT to the Standard Model motivated us to look
for sphericallysymmetric DFT-vacua .
– Such spherically symmetric solutions should admit three
Killing vectors indoubled-yet-gauged spacetime, V Aa , a = 1, 2,
3,
L̂VaHMN = 0 ⇐⇒ (P∇)M (P̄Va)N − (P̄∇)N (PVa)M = 0
L̂Va(e−2d
)= 0 ⇐⇒ ∇M V Ma = 0
which form an so(3) algebra in terms of the C-bracket,
[Va,Vb]C =∑
c�abcVc .
JHP-Rey-Rim-Sakatani 2015
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• The most general spherically symmetric, asymptotically flat
DFT-vacuum solution is
e2φ = γ+(
r−αr+β
) b√a2+b2 + γ−
(r−αr+β
) −b√a2+b2 , B(2) = h cosϑ dt ∧ dϕ ,
ds2 = e2φ[−(
r−αr+β
) a√a2+b2 dt2 +
(r−αr+β
) −a√a2+b2
(dr2 + (r − α)(r + β)dΩ2
)],
where a, b, h (h2 ≤ b2) are three free parameters and
α = aa+b
√a2 + b2 , β = ba+b
√a2 + b2 , γ± = 12
(1±
√1− h2/b2
).
– This is a rederivation of the solution by
Burgess-Myers-Quevedo (1994) whogenerated the above solution by
applying S-duality to the scalar-gravity solutionof Fischer (1948),
Janis-Newman-Winicour (1968). It solves the familiar action,∫
d4x√−|g| e−2φ
(R + 4 |dφ|2 − 112 |dB|
2).
– Equivalently, it solves the EOMs of D = 4 DFT (i.e. pure
Stringy Gravity):
(PABPCD − P̄ABP̄CD)SACBD ≡ 0 , PAC P̄BDSCD ≡ 0 .
– Thus, within the DFT framework, it should be identified as the
DFT-vacuumsolution in analogy with the Schwarzschild solution in
Einstein gravity.
# From GR point of view naked singular, but strictly within DFT
non-singular!
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• The most general spherically symmetric, asymptotically flat
DFT-vacuum solution is
e2φ = γ+(
r−αr+β
) b√a2+b2 + γ−
(r−αr+β
) −b√a2+b2 , B(2) = h cosϑ dt ∧ dϕ ,
ds2 = e2φ[−(
r−αr+β
) a√a2+b2 dt2 +
(r−αr+β
) −a√a2+b2
(dr2 + (r − α)(r + β)dΩ2
)],
where a, b, h (h2 ≤ b2) are three free parameters and
α = aa+b
√a2 + b2 , β = ba+b
√a2 + b2 , γ± = 12
(1±
√1− h2/b2
).
– This is a rederivation of the solution by
Burgess-Myers-Quevedo (1994) whogenerated the above solution by
applying S-duality to the scalar-gravity solutionof Fischer (1948),
Janis-Newman-Winicour (1968). It solves the familiar action,∫
d4x√−|g| e−2φ
(R + 4 |dφ|2 − 112 |dB|
2).
– Equivalently, it solves the EOMs of D = 4 DFT (i.e. pure
Stringy Gravity):
(PABPCD − P̄ABP̄CD)SACBD ≡ 0 , PAC P̄BDSCD ≡ 0 .
– Thus, within the DFT framework, it should be identified as the
DFT-vacuumsolution in analogy with the Schwarzschild solution in
Einstein gravity.
# From GR point of view naked singular, but strictly within DFT
non-singular!
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• The most general spherically symmetric, asymptotically flat
DFT-vacuum solution is
e2φ = γ+(
r−αr+β
) b√a2+b2 + γ−
(r−αr+β
) −b√a2+b2 , B(2) = h cosϑ dt ∧ dϕ ,
ds2 = e2φ[−(
r−αr+β
) a√a2+b2 dt2 +
(r−αr+β
) −a√a2+b2
(dr2 + (r − α)(r + β)dΩ2
)],
where a, b, h (h2 ≤ b2) are three free parameters and
α = aa+b
√a2 + b2 , β = ba+b
√a2 + b2 , γ± = 12
(1±
√1− h2/b2
).
– This is a rederivation of the solution by
Burgess-Myers-Quevedo (1994) whogenerated the above solution by
applying S-duality to the scalar-gravity solutionof Fischer (1948),
Janis-Newman-Winicour (1968). It solves the familiar action,∫
d4x√−|g| e−2φ
(R + 4 |dφ|2 − 112 |dB|
2).
– Equivalently, it solves the EOMs of D = 4 DFT (i.e. pure
Stringy Gravity):
(PABPCD − P̄ABP̄CD)SACBD ≡ 0 , PAC P̄BDSCD ≡ 0 .
– Thus, within the DFT framework, it should be identified as the
DFT-vacuumsolution in analogy with the Schwarzschild solution in
Einstein gravity.
# From GR point of view naked singular, but strictly within DFT
non-singular!
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• The most general spherically symmetric, asymptotically flat
DFT-vacuum solution is
e2φ = γ+(
r−αr+β
) b√a2+b2 + γ−
(r−αr+β
) −b√a2+b2 , B(2) = h cosϑ dt ∧ dϕ ,
ds2 = e2φ[−(
r−αr+β
) a√a2+b2 dt2 +
(r−αr+β
) −a√a2+b2
(dr2 + (r − α)(r + β)dΩ2
)],
where a, b, h (h2 ≤ b2) are three free parameters and
α = aa+b
√a2 + b2 , β = ba+b
√a2 + b2 , γ± = 12
(1±
√1− h2/b2
).
– This is a rederivation of the solution by
Burgess-Myers-Quevedo (1994) whogenerated the above solution by
applying S-duality to the scalar-gravity solutionof Fischer (1948),
Janis-Newman-Winicour (1968). It solves the familiar action,∫
d4x√−|g| e−2φ
(R + 4 |dφ|2 − 112 |dB|
2).
– Equivalently, it solves the EOMs of D = 4 DFT (i.e. pure
Stringy Gravity):
(PABPCD − P̄ABP̄CD)SACBD ≡ 0 , PAC P̄BDSCD ≡ 0 .
– Thus, within the DFT framework, it should be identified as the
DFT-vacuumsolution in analogy with the Schwarzschild solution in
Einstein gravity.
# From GR point of view naked singular, but strictly within DFT
non-singular!
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• We define the ‘proper’ radius,
R :=√
gϑϑ(r) =
[(r − α)(r + β)
(γ+(
r−αr+β
) −a+b√a2+b2 + γ−
(r−αr+β
) −a−b√a2+b2
)] 12
,
which converts the metric into a canonical form where the
angular part is ‘properly’normalized (hence comparable to
observations, e.g. galaxy rotation curves):
ds2 = gttdt2 + gRRdR2 + R2dΩ2 .
• After solving the circular geodesic motion of a point particle
(with the string framemetric), the orbital velocity is given by the
proper radius times the angular velocity,
Vorbit =∣∣∣∣R dϕdt
∣∣∣∣ = [− 12 R dgttdR] 1
2.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• We define the ‘proper’ radius,
R :=√
gϑϑ(r) =
[(r − α)(r + β)
(γ+(
r−αr+β
) −a+b√a2+b2 + γ−
(r−αr+β
) −a−b√a2+b2
)] 12
,
which converts the metric into a canonical form where the
angular part is ‘properly’normalized (hence comparable to
observations, e.g. galaxy rotation curves):
ds2 = gttdt2 + gRRdR2 + R2dΩ2 .
• After solving the circular geodesic motion of a point particle
(with the string framemetric), the orbital velocity is given by the
proper radius times the angular velocity,
Vorbit =∣∣∣∣R dϕdt
∣∣∣∣ = [− 12 R dgttdR] 1
2.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Physical observables of the spherically symmetric DFT-vacuum
• There are three physical observables, on account of the three
free parameters, a, b, h,
M∞G := limR→∞
(RV 2orbit) =12 (a + b
√1− h2/b2) ,
Rphoton = R(rphoton) , rphoton = a + 12(
a−ba+b
)√a2 + b2 ,
QNoether[∂t ] = 14[a +
(a−ba+b
)√a2 + b2
].
– The first defines the asymptotic or Newtonian mass,M∞, from
the Keplerianfall-off of the orbital velocity which eventually
takes place at spatial infinity:
gtt → − 1 + 2M∞GR as R → ∞ .
# Hence, the rotation curve can be non-Keplerian only over a
finite range.Namely, DFT modifies GR at short-distance.
– The second gives the radius of a photon sphere (if
positive).
– The last is the conserved Noether charge for the time
translational symmetry,computable from the DFT-generalization of
the Wald prescription in GR.
JHP-Rey-Rim-Sakatani, c.f. Blair
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Physical observables of the spherically symmetric DFT-vacuum
• There are three physical observables, on account of the three
free parameters, a, b, h,
M∞G := limR→∞
(RV 2orbit) =12 (a + b
√1− h2/b2) ,
Rphoton = R(rphoton) , rphoton = a + 12(
a−ba+b
)√a2 + b2 ,
QNoether[∂t ] = 14[a +
(a−ba+b
)√a2 + b2
].
– The first defines the asymptotic or Newtonian mass,M∞, from
the Keplerianfall-off of the orbital velocity which eventually
takes place at spatial infinity:
gtt → − 1 + 2M∞GR as R → ∞ .
# Hence, the rotation curve can be non-Keplerian only over a
finite range.Namely, DFT modifies GR at short-distance.
– The second gives the radius of a photon sphere (if
positive).
– The last is the conserved Noether charge for the time
translational symmetry,computable from the DFT-generalization of
the Wald prescription in GR.
JHP-Rey-Rim-Sakatani, c.f. Blair
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Various choices of the parameters, a,b,h, (h2 ≤ b2)
• If we set b = h = 0, the solution reduces to the Schwarzschild
metric, with theKeplerian orbital velocity, Vorbit =
√M∞G
R .
• If a = h = 0, we reproduce the renowned orbital velocity
formula by Hernquist,
Rotation curve of Hernquist Model, Vorbit =√
M∞R(R+2M∞G)2
.
Remarkably, the orbital velocity is not monotonic; it assumes
its maximum value,1
2√
2, about 35% of the speed of light, at R = 2M∞G.
• Generically for b 6= 0, rotation curves feature a maximum and
thus non-Keplerian overa finite range, while becoming
asymptotically Keplerian at infinity.
• More interesting limits are the cases of a/b → 0+ or a = 0,
with nontrivial H-flux.JEONG-HYUCK PARK DOUBLED-YET-GAUGED
SPACETIME
-
Various choices of the parameters, a,b,h, (h2 ≤ b2)
• If we set b = h = 0, the solution reduces to the Schwarzschild
metric, with theKeplerian orbital velocity, Vorbit =
√M∞G
R .
• If a = h = 0, we reproduce the renowned orbital velocity
formula by Hernquist,
Rotation curve of Hernquist Model, Vorbit =√
M∞R(R+2M∞G)2
.
Remarkably, the orbital velocity is not monotonic; it assumes
its maximum value,1
2√
2, about 35% of the speed of light, at R = 2M∞G.
• Generically for b 6= 0, rotation curves feature a maximum and
thus non-Keplerian overa finite range, while becoming
asymptotically Keplerian at infinity.
• More interesting limits are the cases of a/b → 0+ or a = 0,
with nontrivial H-flux.JEONG-HYUCK PARK DOUBLED-YET-GAUGED
SPACETIME
-
Various choices of the parameters, a,b,h, (h2 ≤ b2)
• If we set b = h = 0, the solution reduces to the Schwarzschild
metric, with theKeplerian orbital velocity, Vorbit =
√M∞G
R .
• If a = h = 0, we reproduce the renowned orbital velocity
formula by Hernquist,
Rotation curve of Hernquist Model, Vorbit =√
M∞R(R+2M∞G)2
.
Remarkably, the orbital velocity is not monotonic; it assumes
its maximum value,1
2√
2, about 35% of the speed of light, at R = 2M∞G.
• Generically for b 6= 0, rotation curves feature a maximum and
thus non-Keplerian overa finite range, while becoming
asymptotically Keplerian at infinity.
• More interesting limits are the cases of a/b → 0+ or a = 0,
with nontrivial H-flux.JEONG-HYUCK PARK DOUBLED-YET-GAUGED
SPACETIME
-
Various choices of the parameters, a,b,h, (h2 ≤ b2)
• If we set b = h = 0, the solution reduces to the Schwarzschild
metric, with theKeplerian orbital velocity, Vorbit =
√M∞G
R .
• If a = h = 0, we reproduce the renowned orbital velocity
formula by Hernquist,
Rotation curve of Hernquist Model, Vorbit =√
M∞R(R+2M∞G)2
.
Remarkably, the orbital velocity is not monotonic; it assumes
its maximum value,1
2√
2, about 35% of the speed of light, at R = 2M∞G.
• Generically for b 6= 0, rotation curves feature a maximum and
thus non-Keplerian overa finite range, while becoming
asymptotically Keplerian at infinity.
• More interesting limits are the cases of a/b → 0+ or a = 0,
with nontrivial H-flux.JEONG-HYUCK PARK DOUBLED-YET-GAUGED
SPACETIME
-
• By tuning the variable, it is possible to make the maximal
velocity arbitrarily smalland to simulate observed galaxy rotation
curves:
Rotation curves in DFT (dimensionless, nonexhaustive).
– The curves feature a maximum of the orbital velocity after a
fairly rapid rise.It is roughly about 150 km/s c−1 which is
comparable to observations.
– Further, if we let R andM∞ assume the radius and the mass of
the visible matterin the Milky Way, i.e. 15 kpc and 2× 1011M�, we
have as an order of magnitude,R/(M∞G) ' 1.5× 106. This number fits
the scale of the horizontal axis.
– For sufficiently small R/(M∞G), the gravitational force
becomes repulsive.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• By tuning the variable, it is possible to make the maximal
velocity arbitrarily smalland to simulate observed galaxy rotation
curves:
Rotation curves in DFT (dimensionless, nonexhaustive).
– The curves feature a maximum of the orbital velocity after a
fairly rapid rise.It is roughly about 150 km/s c−1 which is
comparable to observations.
– Further, if we let R andM∞ assume the radius and the mass of
the visible matterin the Milky Way, i.e. 15 kpc and 2× 1011M�, we
have as an order of magnitude,R/(M∞G) ' 1.5× 106. This number fits
the scale of the horizontal axis.
– For sufficiently small R/(M∞G), the gravitational force
becomes repulsive.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• By tuning the variable, it is possible to make the maximal
velocity arbitrarily smalland to simulate observed galaxy rotation
curves:
Rotation curves in DFT (dimensionless, nonexhaustive).
– The curves feature a maximum of the orbital velocity after a
fairly rapid rise.It is roughly about 150 km/s c−1 which is
comparable to observations.
– Further, if we let R andM∞ assume the radius and the mass of
the visible matterin the Milky Way, i.e. 15 kpc and 2× 1011M�, we
have as an order of magnitude,R/(M∞G) ' 1.5× 106. This number fits
the scale of the horizontal axis.
– For sufficiently small R/(M∞G), the gravitational force
becomes repulsive.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Discussion
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• We have attempted to view DFT as the stringy extension of, and
hence potentially analternative to, Einstein gravity.
– The fundamental symmetries of DFT unambiguously fix the theory
itself as wellas the couplings to the Standard Model and to a
point-like particle.
• The circular geodesic motion around the most general,
spherically symmetric,asymptotically flat D = 4 DFT-vacuum reveals
that
i) its rotation curve features generically a maximum and thus
non-Keplerian over afinite range (short-distance), while becoming
asymptoticallyKeplerian/Newtonian at infinity (long-distance) as
gtt → − 1 + 2M∞GR .
ii) Furthermore, the gravitational force can be even repulsive
very close to the origin(far-short-distance).
• DFT is, by nature, Stringy Gravity, which is compatible with
GR: it still includes GR.
– Yet, the self-interaction of the massless NS-NS sector can
‘modify’ GR.– From the conventional GR point of view, the scalar
dilaton and the B-field may
well be regarded as ‘dark matter’ (c.f. axion) or ‘dark
graviy’.
• Deeper understanding of the three free parameters of the
DFT-vacuum, perhaps as theintrinsic properties of matter or an
elementary particle, would be desirable.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• We have attempted to view DFT as the stringy extension of, and
hence potentially analternative to, Einstein gravity.
– The fundamental symmetries of DFT unambiguously fix the theory
itself as wellas the couplings to the Standard Model and to a
point-like particle.
• The circular geodesic motion around the most general,
spherically symmetric,asymptotically flat D = 4 DFT-vacuum reveals
that
i) its rotation curve features generically a maximum and thus
non-Keplerian over afinite range (short-distance), while becoming
asymptoticallyKeplerian/Newtonian at infinity (long-distance) as
gtt → − 1 + 2M∞GR .
ii) Furthermore, the gravitational force can be even repulsive
very close to the origin(far-short-distance).
• DFT is, by nature, Stringy Gravity, which is compatible with
GR: it still includes GR.
– Yet, the self-interaction of the massless NS-NS sector can
‘modify’ GR.– From the conventional GR point of view, the scalar
dilaton and the B-field may
well be regarded as ‘dark matter’ (c.f. axion) or ‘dark
graviy’.
• Deeper understanding of the three free parameters of the
DFT-vacuum, perhaps as theintrinsic properties of matter or an
elementary particle, would be desirable.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• We have attempted to view DFT as the stringy extension of, and
hence potentially analternative to, Einstein gravity.
– The fundamental symmetries of DFT unambiguously fix the theory
itself as wellas the couplings to the Standard Model and to a
point-like particle.
• The circular geodesic motion around the most general,
spherically symmetric,asymptotically flat D = 4 DFT-vacuum reveals
that
i) its rotation curve features generically a maximum and thus
non-Keplerian over afinite range (short-distance), while becoming
asymptoticallyKeplerian/Newtonian at infinity (long-distance) as
gtt → − 1 + 2M∞GR .
ii) Furthermore, the gravitational force can be even repulsive
very close to the origin(far-short-distance).
• DFT is, by nature, Stringy Gravity, which is compatible with
GR: it still includes GR.
– Yet, the self-interaction of the massless NS-NS sector can
‘modify’ GR.– From the conventional GR point of view, the scalar
dilaton and the B-field may
well be regarded as ‘dark matter’ (c.f. axion) or ‘dark
graviy’.
• Deeper understanding of the three free parameters of the
DFT-vacuum, perhaps as theintrinsic properties of matter or an
elementary particle, would be desirable.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• We have attempted to view DFT as the stringy extension of, and
hence potentially analternative to, Einstein gravity.
– The fundamental symmetries of DFT unambiguously fix the theory
itself as wellas the couplings to the Standard Model and to a
point-like particle.
• The circular geodesic motion around the most general,
spherically symmetric,asymptotically flat D = 4 DFT-vacuum reveals
that
i) its rotation curve features generically a maximum and thus
non-Keplerian over afinite range (short-distance), while becoming
asymptoticallyKeplerian/Newtonian at infinity (long-distance) as
gtt → − 1 + 2M∞GR .
ii) Furthermore, the gravitational force can be even repulsive
very close to the origin(far-short-distance).
• DFT is, by nature, Stringy Gravity, which is compatible with
GR: it still includes GR.
– Yet, the self-interaction of the massless NS-NS sector can
‘modify’ GR.– From the conventional GR point of view, the scalar
dilaton and the B-field may
well be regarded as ‘dark matter’ (c.f. axion) or ‘dark
graviy’.
• Deeper understanding of the three free parameters of the
DFT-vacuum, perhaps as theintrinsic properties of matter or an
elementary particle, would be desirable.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• While the proper radius, R, is the dimensionful physical
radius,the normalized radius, R/(M∞G), is the mathematically
natural dimensionlessvariable which essentially probes the
theoretical nature of the gravitational force.
– Intriguingly, R/(M∞G) is thousand times smaller for the Milky
Way comparedto the Earth at each surface (of the visible matter):
1.5× 106 versus 1.4× 109.
– Generically, if the mass density is constant, R/(M∞G) becomes
smaller as thephysical radius, R, grows.
Cosmic Uroboros:
– The observations of stars and galaxies faraway, or the dark
matter and the dark energyproblems, are revealing the
short-distance natureof gravity!
– The repulsive gravitational force at very short-distance,
R/(M∞G) → 0+, may be responsiblefor the acceleration of the
Universe.
Thank you.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• While the proper radius, R, is the dimensionful physical
radius,the normalized radius, R/(M∞G), is the mathematically
natural dimensionlessvariable which essentially probes the
theoretical nature of the gravitational force.
– Intriguingly, R/(M∞G) is thousand times smaller for the Milky
Way comparedto the Earth at each surface (of the visible matter):
1.5× 106 versus 1.4× 109.
– Generically, if the mass density is constant, R/(M∞G) becomes
smaller as thephysical radius, R, grows.
Cosmic Uroboros:
– The observations of stars and galaxies faraway, or the dark
matter and the dark energyproblems, are revealing the
short-distance natureof gravity!
– The repulsive gravitational force at very short-distance,
R/(M∞G) → 0+, may be responsiblefor the acceleration of the
Universe.
Thank you.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• While the proper radius, R, is the dimensionful physical
radius,the normalized radius, R/(M∞G), is the mathematically
natural dimensionlessvariable which essentially probes the
theoretical nature of the gravitational force.
– Intriguingly, R/(M∞G) is thousand times smaller for the Milky
Way comparedto the Earth at each surface (of the visible matter):
1.5× 106 versus 1.4× 109.
– Generically, if the mass density is constant, R/(M∞G) becomes
smaller as thephysical radius, R, grows.
Cosmic Uroboros:
– The observations of stars and galaxies faraway, or the dark
matter and the dark energyproblems, are revealing the
short-distance natureof gravity!
– The repulsive gravitational force at very short-distance,
R/(M∞G) → 0+, may be responsiblefor the acceleration of the
Universe.
Thank you.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• While the proper radius, R, is the dimensionful physical
radius,the normalized radius, R/(M∞G), is the mathematically
natural dimensionlessvariable which essentially probes the
theoretical nature of the gravitational force.
– Intriguingly, R/(M∞G) is thousand times smaller for the Milky
Way comparedto the Earth at each surface (of the visible matter):
1.5× 106 versus 1.4× 109.
– Generically, if the mass density is constant, R/(M∞G) becomes
smaller as thephysical radius, R, grows.
Cosmic Uroboros:
– The observations of stars and galaxies faraway, or the dark
matter and the dark energyproblems, are revealing the
short-distance natureof gravity!
– The repulsive gravitational force at very short-distance,
R/(M∞G) → 0+, may be responsiblefor the acceleration of the
Universe.
Thank you.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• While the proper radius, R, is the dimensionful physical
radius,the normalized radius, R/(M∞G), is the mathematically
natural dimensionlessvariable which essentially probes the
theoretical nature of the gravitational force.
– Intriguingly, R/(M∞G) is thousand times smaller for the Milky
Way comparedto the Earth at each surface (of the visible matter):
1.5× 106 versus 1.4× 109.
– Generically, if the mass density is constant, R/(M∞G) becomes
smaller as thephysical radius, R, grows.
Cosmic Uroboros:
– The observations of stars and galaxies faraway, or the dark
matter and the dark energyproblems, are revealing the
short-distance natureof gravity!
– The repulsive gravitational force at very short-distance,
R/(M∞G) → 0+, may be responsiblefor the acceleration of the
Universe.
Thank you.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
• While the proper radius, R, is the dimensionful physical
radius,the normalized radius, R/(M∞G), is the mathematically
natural dimensionlessvariable which essentially probes the
theoretical nature of the gravitational force.
– Intriguingly, R/(M∞G) is thousand times smaller for the Milky
Way comparedto the Earth at each surface (of the visible matter):
1.5× 106 versus 1.4× 109.
– Generically, if the mass density is constant, R/(M∞G) becomes
smaller as thephysical radius, R, grows.
Cosmic Uroboros:
– The observations of stars and galaxies faraway, or the dark
matter and the dark energyproblems, are revealing the
short-distance natureof gravity!
– The repulsive gravitational force at very short-distance,
R/(M∞G) → 0+, may be responsiblefor the acceleration of the
Universe.
Thank you.
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
Talk based on collaborations with Imtak Jeon (8 papers),
Kanghoon Lee (8 papers), Yoonji Suh (3 papers),Chris Blair, Emanuel
Malek, Wonyoung Cho, Jose Fernández-Melgarejo, Soo-Jong Rey,
Woohyun Rim, Yuho Sakatani,Sung Moon Ko, Charles Melby-Thompson,
Rene Meyér, Minwoo Suh, Kang-Sin Choi and Xavier Bekaert.
• Differential geometry with a projection: Application to double
field theory 1011.1324 JHEP
• Stringy differential geometry, beyond Riemann 1105.6294
PRD
• Incorporation of fermions into double field theory 1109.2035
JHEP
• Ramond-Ramond Cohomology and O(D,D) T-duality 1206.3478
JHEP
• Supersymmetric Double Field Theory: Stringy Reformulation of
Supergravity 1112.0069 PRD
• Stringy Unification of IIA and IIB Supergravities underN= 2 D=
10 Supersymmetric Double Field Theory1210.5078 PLB
• Supersymmetric gauged Double Field Theory: Systematic
derivation by virtue of ‘Twist’ 1505.01301 JHEP
• Comments on double field theory and diffeomorphisms 1304.5946
JHEP
• Covariant action for a string in doubled yet gauged spacetime
1307.8377 NPB
• Green-Schwarz superstring on doubled-yet-gauged spacetime
1609.04265 JHEP
• Double field formulation of Yang-Mills theory 1102.0419
PLB
• Standard Model as a Double Field Theory 1506.05277 PRL
• The rotation curve of a point particle in stringy gravity
1606.09307
• O(D,D) Covariant Noether Currents and Global Charges in Double
Field Theory 1507.07545 JHEP
• Dynamics of Perturbations in Double Field Theory &
Non-Relativistic String Theory 1508.01121 JHEP
• Higher Spin Double Field Theory: A Proposal 1605.00403
JHEP
• U-geometry: SL(5) ⇒ U-gravity: SL(N) 1302.1652 JHEP/1402.5027
JHEP
• M-theory and Type IIB from a Duality Manifest Action 1311.5109
JHEP
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME
-
The End
JEONG-HYUCK PARK DOUBLED-YET-GAUGED SPACETIME